Express vectors in different coordinate systems

I don't understand this problem. This is supposedly a review question from calc 3, but I've never seen anything like this before. Fortunately, I solved it, but I just don't understand what I did. (It's probably the weird notation I'm not used to.)

Express the r-component, , or a vector A at

(A) In terms of and in Cartesian coordinates, and
(B) in terms of and in spherical coordinates.

My book says:
"The vector in Cartesian coordinates with components can be written as:

Answers:
(A)
(B)

So for part (A), is that written in the same form as the unit vectors i hat, j hat, and k hat? And for part (B), why are both components divided by the radius?